At the school close to Quiz Master Shop Towers they have four houses, and the school recently ran it's sports day.
There were five races each with one competitor from each of the four houses. By an odd coincidence there was one runner named Smith in each of the five races. We were talking to Mr Smith the local bookmaker before the races; he is the parent of one of the children running. We asked him what odds he would offer that at least one of the Smiths won a race.
He replied that he knew nothing of the running abilities of any of the 20 competitors, but he would offer odds of 4 to 1 on.
Assuming that in each race all the runners have an equal chance of winning, is the a generous offer?
The probability of a Smith not winning the first race is 3/4 or 75%.
The probability of a Smith not winning the second race is also 3/4 or 75%.
So the probability of a Smith not winning either of the first two races is 3/4 x 3/4 or 9/16.
By extension the probability of a Smith winning none of the five races is 3/4 x 3/4 x 3/4 x 3/4 x 3/4 or 243/1024.
Thus the probability of a Smith winning at least one race is (1024 - 243)/1024 which is 781/1024 or just over 76%.
Odds of 4 to 1 on means 80%, so not very generous.